How representative are Australia’s elected representatives of the population as a whole? There is a clear disparity between the support of a party, in terms of popular vote, and seats won, in the results of the 2016 federal election.

How many voters do the major parties represent?

As of the morning of July 12, the figures in parentheses in the table below are the number of seats won out of 150 in the House of Representatives, and out of 76 in the Senate, in the 2016 election:

There are two points to note from this table.

First, the percentages for each party are closer in the Senate than they are in the House of Representatives.

Second, the major parties fared much better in the House of Representatives than the Greens.

The first point is due to proportional representation being used to elect the Senate. If a party receives X% of the vote in a state, it should receive “roughly” X% of the senators. But we are sweeping an awful lot under the rug with the word “roughly”. And the national vote obscures the differences in the states’ population: Tasmania elects the same number of senators as New South Wales, even though the population of NSW is more than 12 times that of Tasmania.

More senators means more representation

At a normal half-Senate election, where each state elects six senators, the quota is one-seventh of the total vote, or 14.3%. If candidates receive more than one quota they are elected. If, after others are eliminated, a candidate has only 14.2% of the vote, they miss out.

At a double-dissolution election the quota is almost halved to a 13th of the total vote, or 7.7%. Therefore, a candidate will miss out if they are the last one standing with only 7.6% of the vote.

In other words, in a normal election it is “unfair” that candidates can be excluded even with 14.2% of the vote. The situation is much “fairer” in a double-dissolution election; only those with 7.6% or less will miss out.

The situation would get fairer still if more senators were up for election. If, say, there were 100 senators, then only those with less than 1% of the vote would miss out.

Even though increasing the number of senators is possible without a referendum, and even though parliament has done this three times (in 1948, 1975, and 1984), there appears to be no push for an increase at present.

Preferences and spread of vote

The second point is due to the House of Representatives’ structure: that is, one member is elected in each electorate. If Party X received 10% of the first-preference vote in each electorate, it would be almost impossible for any candidates from the party to be elected.

The more likely scenario is that, at some point in the vote count, candidates from Party X would have the fewest votes of all remaining candidates and hence be eliminated.

First-preference votes are not everything, however. The final votes after preferences are those that determine the election. Nevertheless, at least intuitively, the greater the nationwide primary vote for Party X, the greater the number of seats for Party X. But there are two obvious caveats.

Results could be skewed between electorates. For example, the Labor vs Coalition vote could be 10:90 and 60:40 in two electorates. Labor would then have only 35% of the total vote (half of 10% and half of 60%), but win as many seats as the Coalition (one each).

Preferences can get parties over the line. For example, the Labor vs Coalition vs Greens vote could be 40:45:15 in a particular electorate. The Greens candidate, having the fewest votes, would be eliminated. If more than two-thirds of the Greens voters have the Labor candidate as a second preference, the Labor candidate would be elected, since 40 + (2/3 x 15) is 50.

Votes or seats, what’s more important?

Below is a graph of first-preference votes compared to House of Representative seats won in all federal elections since 1949. Labor is represented by the red dots; the Coalition by the blue.

Here we see a definite trend, best approximated by the blue diagonal line drawn through the points, showing an increase in seats won when the primary vote increases.

When a point lies above this blue diagonal line the party is “lucky” – that is, it won more seats than it “should have won”. This is due to a combination of skewed results across electorates and favourable preferences.

Points below the blue diagonal line are unlucky: the spread of the national vote and preferences conspired against the party.

The second red dot from the left is Labor’s primary vote in the 2016 election. This is claimed to be too weak to be seriously competitive.

This is borne out by the large distance of the dot above the diagonal blue line: Labor was lucky to win the number of seats it did. We can also use the graph to solve for the votes required to win more than half of the seats – this turns out to be 45%.

Therefore, if a party won 45% of the vote, it would expect to form government. The vertical red line represents this cut-off.

However, the horizontal orange line represents the cut-off of the elections where the parties had enough seats to form government, and below the line when they did not.

If a party won less than 45%, it could get “lucky” and still form government, as did six dots contained in the region above the orange line and to the left of the red line.

Opposite these points (below orange and right of red) are the unlucky points: parties securing more votes than they “needed” but still failing to form government. The Coalition has been lucky more times than Labor.

For the House of Representatives, each electorate chooses one person to represent it; for the Senate, each state chooses six people to represent it. This leads to a “fairer” representation in the Senate, at least with respect to the proportion of votes cast and seats won. Having even more senators would make this even fairer.